We first consider the semidirect product relative Rota-Baxter algebra \(M \oplus N \xrightarrow{} A \oplus B\)  given in Proposition REF . Hence it follows from Proposition REF  that the space \(M \oplus N\)  carries a dendriform algebra structure with products
\((m,n) \prec _{R \oplus S} (m^{\prime }, n^{\prime }) =~& (m \cdot _M R (m^{\prime }), ~ l (m, S (n^{\prime }) + n \cdot _N R (m^{\prime })),\\(m,n) \succ _{R \oplus S} (m^{\prime }, n^{\prime }) =~& (R(m) \cdot _M m^{\prime }, ~ R(m) \cdot _N n^{\prime } + r (S(n), m^{\prime })).\) 

The corresponding total associative multiplication is given by
\((m,n) \ast _{R \oplus S} (m^{\prime },n^{\prime }) = (m \ast _R m^{\prime }, ~ l (m, S (n^{\prime }) + n \cdot _N R (m^{\prime }) + R(m) \cdot _N n^{\prime } + r (S(n), m^{\prime })).\) 

Moreover, \(A \oplus B\)  is a bimodule over this associative algebra with left and right actions
\((m,n) \triangleright _{A \oplus B} (a, b) =~& \big ( R(m) \cdot a - R (m \cdot _M a),~ R(m) \cdot _B b + S(n) \cdot _B a - S ( l (m, b) + n \cdot _N a) \big ),\\(a,b) \triangleleft _{A \oplus B} (m,n) =~& \big ( a \cdot R(m) - R (a \cdot _M m), ~ a \cdot _B S(n) + b \cdot _B R(m) - S ( a \cdot _N n + r (b, m)) \big ).\) 

It follows from [1]} that the collection \(\lbrace \Psi _k\rbrace _{k \ge 1}\)  of maps
\(& \Psi _k : C^k_\mathsf {H} (M \oplus N, A \oplus B) \rightarrow C^{k+1}_\mathsf {D} (M \oplus N, M \oplus N) \\\Psi _k (F)([r]; (m_1, n_1),& \ldots , (m_{k+1}, n_{k+1})) = {\left\lbrace \begin{array}{ll}(-1)^{k+1} ~ (m_1, n_1) \triangleleft F ( (m_2, n_2), \ldots , (m_{k+1}, n_{k+1})) & \text{if } r = 1 \\0 & \text{if } 2 \le r \le k \\F ( (m_1, n_1), \ldots , (m_{k}, n_{k})) \triangleright (m_{k+1}, n_{k+1}) & \text{if } r = k+1\end{array}\right.}\) 

defines a morphism of cochain complexes from \(\lbrace  C^\bullet _\mathsf {H} (M \oplus N, A \oplus B), \delta _{ M \oplus N, A \oplus B } \rbrace \)  to \(\lbrace  C^{\bullet + 1}_\mathsf {D} (M \oplus N, M \oplus N), \delta _\mathsf {D} \rbrace \) . Note that \(\lbrace  C^\bullet _\mathsf {H} (M_\mathrm {Tot}, B_{\triangleright , \triangleleft }), \delta _{M,B} \rbrace \)  is a subcomplex of \(\lbrace  C^\bullet _\mathsf {H} (M \oplus N, A \oplus B), \delta _{M\oplus N, A \oplus B} \rbrace \) , and \(\lbrace C^{\bullet +1}_\mathsf {D} (M,N), \delta _\mathsf {D} \rbrace \)  is a subcomplex of \(\lbrace  C^{\bullet + 1}_\mathsf {D} (M \oplus N, M \oplus N), \delta _\mathsf {D} \rbrace \) . Finally, the result follows as the restriction of \(\Psi _k\)  to the subspace \(C^k_\mathsf {H} (M_\mathrm {Tot}, B_{\triangleright , \triangleleft })\)  is given by \(\psi _k\) , for all \(k \ge 1\) . This proves the result.
